U.S. Pat. No. 4,070,136 discloses a conventional swash plate type compressor having a pair of horizontal axially aligned cylinder blocks which form a combined cylinder. Inside the combined cylinder are formed axially extending cylinder bores, and the combined cylinder is closed at both ends by front and rear housings, via valve plates. Centrally passing through the combined block, a drive shaft is rotatably supported by a suitable bearing means. To the middle of the drive shaft is fixed a swash plate operatively connected to, via ball bearings and shoes, double headed pistons slidably fitted in the cylinder bores. Thus, the rotating of the swash plate causes the reciprocal compressing motion of the pistons within the cylinder bores. The front and rear housings have refrigerant suction chambers and refrigerant discharge chambers, which are interconnected with the cylinder bores and are connectable to an outside air conditioning system by means of appropriate refrigerant flow pipelines. The above-mentioned conventional compressor has therein such an internal arrangement that while the double headed pistons reciprocate in the cylinder bores, the rotating motion of each double headed piston about its own axis is stopped by the swash plate. More specifically, the double headed pistons are arranged adjacent to the radially outward part of the swash plate with respect to the axis of rotation of the swash plate. Therefore, if each double headed piston rotates about its own axis, a part of each piston abuts against the cylindrical outer surface of the swash plate. As a result, a further rotation of each piston is prevented by the cylindrical outer surface of the swash plate. At this stage, a very small amount of space is provided between each piston and the cylindrical outer surface of the swash plate so that the generation of noisy sounds by the above-mentioned abuting motion of each piston is limited to the lowest possible level. However, the very small amount of space provided between each piston and the cylindrical outer surface of the swash plate has shown diverse defects, as described later with reference to the accompanying drawings, from the point of view of performance and durability of the swash plate type compressor.